Weierstrass Substitution

From ProofWiki
Jump to navigation Jump to search


This article has been identified as a candidate for Featured Proof status.
If you do not believe that this proof is worthy of being a Featured Proof, please state your reasons on the talk page.
To discuss this page in more detail, feel free to use the talk page.


Proof Technique

The Weierstrass substitution is an application of Integration by Substitution.


The substitution is:

$u \leftrightarrow \tan \dfrac \theta 2$

for $-\pi < \theta < \pi$, $u \in \R$.


It yields:

Sine

$\sin \theta = \dfrac {2 u} {1 + u^2}$


Cosine

$\cos \theta = \dfrac {1 - u^2} {1 + u^2}$


Derivative

$\dfrac {\d \theta} {\d u} = \dfrac 2 {1 + u^2}$


The above results can be stated:

$\ds \int \map F {\sin \theta, \cos \theta} \rd \theta = 2 \int \map F {\frac {2 u} {1 + u^2}, \frac {1 - u^2} {1 + u^2} } \frac {d u} {1 + u^2}$

where $u = \tan \dfrac \theta 2$.


Also known as

The technique of Weierstrass Substitution is also known as Tangent Half-Angle Substitution.

Some sources call these results the Tangent-of-Half-Angle Formulae.

Other sources refer to them merely as the Half-Angle Formulas or Half-Angle Formulae.


Also see


Source of Name

This entry was named for Karl Theodor Wilhelm Weierstrass.


Sources

Retrieved from "https://proofwiki.org/w/index.php?title=Weierstrass_Substitution&oldid=740339"